Theorem hbsbcgd 2517

Description: Deduction version of hbsbcg 2466. (The proof was shortened by Andrew Salmon, 21-Jun-2011.)

Hypotheses

Ref	Expression
hbsbcgd.1	|- (ph -> A.xph)
hbsbcgd.2	|- (ph -> A.yph)
hbsbcgd.3	|- (ph -> (z e. A -> A.x z e. A))
hbsbcgd.4	|- (ph -> (ps -> A.xps))

Assertion

Ref	Expression
hbsbcgd	|- ((ph /\ A e. C) -> ([A / y]ps -> A.x[A / y]ps))

Distinct variable groups:   z,A   ph,z   x,z

Proof of Theorem hbsbcgd

Step	Hyp	Ref	Expression
1		hbsbcgd.3	. . . . . . 7 |- (ph -> (z e. A -> A.x z e. A))
2	1	19.21aiv 1664	. . . . . 6 |- (ph -> A.z(z e. A -> A.x z e. A))
3		abidhb 2423	. . . . . 6 |- (A.z(z e. A -> A.x z e. A) -> {w | A.x w e. A} = A)
4		eleq1 1957	. . . . . 6 |- ({w | A.x w e. A} = A -> ({w | A.x w e. A} e. _V <-> A e. _V))
5	2, 3, 4	3syl 24	. . . . 5 |- (ph -> ({w | A.x w e. A} e. _V <-> A e. _V))
6	5	biimpar 461	. . . 4 |- ((ph /\ A e. _V) -> {w | A.x w e. A} e. _V)
7		hba1 1350	. . . . . 6 |- (A.x w e. A -> A.xA.x w e. A)
8	7	hbab 1875	. . . . 5 |- (z e. {w | A.x w e. A} -> A.x z e. {w | A.x w e. A})
9		hba1 1350	. . . . 5 |- (A.xps -> A.xA.xps)
10	8, 9	hbsbcg 2466	. . . 4 |- ({w | A.x w e. A} e. _V -> ([{w | A.x w e. A} / y]A.xps -> A.x[{w | A.x w e. A} / y]A.xps))
11	6, 10	syl 12	. . 3 |- ((ph /\ A e. _V) -> ([{w | A.x w e. A} / y]A.xps -> A.x[{w | A.x w e. A} / y]A.xps))
12		dfsbcq 2455	. . . . . 6 |- ({w | A.x w e. A} = A -> ([{w | A.x w e. A} / y]A.xps <-> [A / y]A.xps))
13	2, 3, 12	3syl 24	. . . . 5 |- (ph -> ([{w | A.x w e. A} / y]A.xps <-> [A / y]A.xps))
14	13	adantr 425	. . . 4 |- ((ph /\ A e. _V) -> ([{w | A.x w e. A} / y]A.xps <-> [A / y]A.xps))
15		hbsbcgd.2	. . . . 5 |- (ph -> A.yph)
16		ax-4 1319	. . . . . 6 |- (A.xps -> ps)
17		hbsbcgd.4	. . . . . 6 |- (ph -> (ps -> A.xps))
18	16, 17	impbid2 576	. . . . 5 |- (ph -> (A.xps <-> ps))
19	15, 18	sbcbid 2504	. . . 4 |- ((ph /\ A e. _V) -> ([A / y]A.xps <-> [A / y]ps))
20	14, 19	bitrd 587	. . 3 |- ((ph /\ A e. _V) -> ([{w | A.x w e. A} / y]A.xps <-> [A / y]ps))
21		hbsbcgd.1	. . . . . . 7 |- (ph -> A.xph)
22	21	a1d 15	. . . . . 6 |- (ph -> (ph -> A.xph))
23		ax-17 1317	. . . . . . . 8 |- (z e. _V -> A.x z e. _V)
24	23	a1i 8	. . . . . . 7 |- (ph -> (z e. _V -> A.x z e. _V))
25	21, 1, 24	hbeld 2425	. . . . . 6 |- (ph -> (A e. _V -> A.x A e. _V))
26	22, 25	hband 1469	. . . . 5 |- (ph -> ((ph /\ A e. _V) -> A.x(ph /\ A e. _V)))
27	26	anabsi5 553	. . . 4 |- ((ph /\ A e. _V) -> A.x(ph /\ A e. _V))
28	27, 20	albid 1459	. . 3 |- ((ph /\ A e. _V) -> (A.x[{w | A.x w e. A} / y]A.xps <-> A.x[A / y]ps))
29	11, 20, 28	3imtr3d 601	. 2 |- ((ph /\ A e. _V) -> ([A / y]ps -> A.x[A / y]ps))
30		elisset 2299	. 2 |- (A e. C -> A e. _V)
31	29, 30	sylan2 500	1 |- ((ph /\ A e. C) -> ([A / y]ps -> A.x[A / y]ps))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  [wsbc 1534  {cab 1871  _Vcvv 2292

This theorem is referenced by:  hbcsbgd 2571

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454

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